Let X 1 , X 2 denote positive heavy-tailed random variables with continuous marginal distribution functions F 1 and F 2 , respectively. The asymptotic behavior of the tail of X 1 + X 2 is studied in a general copula framework and some bounds and extremal properties are provided. For more specific assumptions on F 1 , F 2 and the underlying dependence structure of X 1 and X 2 , we survey explicit asymptotic results available in the literature and add several new cases.
In this paper we extend some results about the probability that the sum of n dependent subexponential random variables exceeds a given threshold u. In particular, the case of non-identically distributed and not necessarily positive random variables is investigated. Furthermore we establish criteria how far the tail of the marginal distribution of an individual summand may deviate from the others so that it still influences the asymptotic behavior of the sum. Finally we explicitly construct a dependence structure for which, even for regularly varying marginal distributions, no asymptotic limit of the tail sum exists. Some explicit calculations for diagonal copulas and t-copulas are given.
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